Thermal convection of a viscoelastic fluid in an open-top porous layer heated from below

Buoyancy-driven convection of a viscoelastic fluid saturated in an open-top porous square box is studied based on a modified Darcy's law. The results are compared with those for a Newtonian fluid under the same boundary conditions and those for the viscoelastic fluid under a closed-top boundary. In particular, the critical Darcy–Rayleigh number Ra for onset of convection is determined first by using the linear stability theory. Then the effects of the relaxation time and the retardation time of the viscoelastic fluid on the heat transfer rate and the flow pattern are investigated numerically. The results reveal some interesting properties of thermal convection for the viscoelastic fluid. The relaxation time makes the fluid easier to destabilize while the retardation time tends to stabilize the fluid motion in the porous medium, and larger heat transfer rate can be achieved with larger value of the relaxation time and decreased retardation time. Furthermore, larger relaxation time facilitates earlier bifurcation of the flow pattern as Ra increases, but bifurcation can be postponed with increased retardation time. For larger ratio of relaxation time over retardation time, the flow pattern is more complicated and the frequency of flow oscillation also increases. Finally, large ratio of relaxation time over retardation time can make the open-top boundary impermeable due to the viscoelastic effect on the fluid.     

Electrohydrodynamic peristaltic flow of a dielectric Oldroydian viscoelastic fluid in a flexible channel with heat transfer

This paper discusses the effects of a vertical a.c. electric field and heat transfer on a peristaltic flow of an incompressible dielectric viscoelastic fluid in a symmetric flexible channel. The mathematical modeling includes interactions among the electric field, flow field, and temperature. The perturbation solution of the modeled problem is derived by considering a small wave number. The influence of pertinent parameters is demonstrated and discussed. The numerical results show that the possibility of flow reversal increases near the lower bound of the channel and decreases near the upper bound of the channel as the electrical Rayleigh number, the Reynolds number, and the Weissenberg number increase, whereas the opposite effect is observed as the temperature parameter and the Weissenberg number increase. It is observed that the size of the trapped bolus decreases at the upper bound of the channel and increases at the lower bound of the channel with increasing electrical Rayleigh number, whereas the opposite effect is observed as the temperature parameter increases. The results also show that the trapped bolus in the case of an Oldroydian fluid is smaller than that for a Newtonian fluid.     

Onset of electroconvective instability of Oldroydian viscoelastic liquid layer in Brinkman porous medium

The problem of the onset of electrohydrodynamic instability in a horizontal layer of Oldroydian viscoelastic dielectric liquid through Brinkman porous medium under the simultaneous action of a certical ac electric field and a vertical temperature gradient is analyzed. Applying linear stability theory, we derive an equation of eight order. Under somewhat suitable boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Both the cases of stationary and oscillatory instabilities are discussed if the liquid layer is heated from below or above. The effects of the porosity of porous medium, the medium permeability, the Prandtl number, the ratio of retardation time to relaxation time, the elastic number, in the presence or absence of Rayleigh number are shown graphically for both cases. Some of the known results are derived as special cases. The electrical force has been shown to be the sole agency causing instability of the considered system since it is much more important than the buoyancy force even if the medium is porous.     

Determining the shear modulus of water in experiments with a floating disk

The problem of decaying rotation of a disk floating on the surface of a viscoelastic fluid in a cylindrical container is solved by numerical methods. The motion is found to have the form of decaying oscillations observed previously for water. In addition to the viscosity coefficient, the constructed mathematical model of the viscoelastic fluid has two more independent parameters: shear modulus and time of relaxation of elastic stresses. Elastic parameters of water are determined through comparisons with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 100–103, January–February, 2008.     

The Flow Analysis of Viscoelastic Fluid with Fractional Order Derivative in Horizontal Well

The fractional calculus approach is introduced into the seepage mechanics. A three-dimensional relaxation model of viscoelastic fluid is built. The models based on four boundary conditions of exact solution in Laplace space for some unsteady flows in an infinite reservoir is obtained by using the Laplace transform and Fourier sine and cosine integral transform. The pressure transient behavior of non-Newtonian viscoelastic fluid is studied by using Stehfest method of the numerical Laplace transform inversion and Gauss–Laguerre numerical integral formulae. The viscoelastic fluid is very sensitive to the order of the fractional derivative. The change rules of pressure are discussed when the parameters of the models change. The plots of type pressure curves are given, and the results can be provided to theoretical basis and well-test method for oil field.     

From dynamic modulus via different relaxation spectra to relaxation and creep functions

The main goal of the paper is to compare predictive power of relaxation spectra found by different methods of calculations. The experimental data were obtained for a new family of propylene random copolymers with 1-pentene as a comonomer. The results of measurements include flow curves, viscoelastic properties, creep curves and rubbery elasticity of copolymer melts. Different relaxation spectra were calculated using independent methods based on different ideas. It lead to various distributions of relaxation times and their “weights”. However, all of them correctly describe the frequency dependencies of dynamic modulus. Besides, calculated spectra were used for finding integral characteristics of viscoelastic behaviour of a material (Newtonian viscosity, the normal stress coefficient, steady-state compliance). In this sense all approaches are equivalent, though it appears impossible to estimate instantaneous modulus. The most crucial arguments in estimating the results of different approaches is calculating the other viscoelastic function and predicting behaviour of a material in various deformation modes. It is the relaxation and creep functions. The results of relaxation curve calculations show that all methods used give rather similar results in the central part of the curves, but the relaxation curves begin to diverge when approaching the high-time (low-frequency) boundary of the relaxation curves. The distributions of retardation times calculated through different approaches also appear very different. Meanwhile, predictions of the creep curves based on these different retardation spectra are rather close to each other and coincide with the experimental points in the wide time range. Relatively slight divergences are observed close to the upper boundary of the experimental window. All these results support the conclusion about a rather free choice of the relaxation time spectrum in fitting experimental data and predicting viscoelastic behaviour of a material in different deformation modes. Received: 15 March 2000 Accepted: 18 September 2000     

Wave resistance of an air-cushion vehicle in unsteady motion over an ice sheet

Unsteady rectilinear motion of an air-cushion vehicle over an ice sheet at various speeds is considered. Ice is modeled by a viscoelastic ice plate. The effects of the basin depth, the thickness and relaxation time of ice, vehicle length, acceleration, deceleration, and speed of uniform motion on the wave resistance of the vehicle are analyzed. Maneuvering methods for increasing or lowering the wave resistance of the vehicle are proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 89–99, January–February, 2008.     

Electroosmosis modulated peristaltic biorheological flow through an asymmetric microchannel: mathematical model

A theoretical study is presented of peristaltic hydrodynamics of an aqueous electrolytic non-Newtonian Jeffrey bio-rheological fluid through an asymmetric microchannel under an applied axial electric field. An analytical approach is adopted to obtain the closed form solution for velocity, volumetric flow, pressure difference and stream function. The analysis is also restricted under the low Reynolds number assumption (Stokes flow) and lubrication theory approximations (large wavelength). Small ionic Peclét number and Debye–Hückel linearization (i.e. wall zeta potential ≤ 25 mV) are also considered to simplify the Nernst–Planck and Poisson–Boltzmann equations. Streamline plots are also presented for the different electro-osmotic parameter, varying magnitudes of the electric field (both aiding and opposing cases) and for different values of the ratio of relaxation to retardation time parameter. Comparisons are also included between the Newtonian and general non-Newtonian Jeffrey fluid cases. The results presented here may be of fundamental interest towards designing lab-on-a-chip devices for flow mixing, cell manipulation, micro-scale pumps etc. Trapping is shown to be more sensitive to an electric field (aiding, opposing and neutral) rather than the electro-osmotic parameter and viscoelastic relaxation to retardation ratio parameter. The results may also help towards the design of organ-on-a-chip like devices for better drug design.     

Strain recovery of model immiscible blends without compatibilizer

Strain recovery after the cessation of shear was studied in model immiscible blends composed of polyisobutylene drops (10–30% by weight) in a polydimethylsiloxane matrix. Blends of viscosity ratio (viscosity of the drops relative to the matrix viscosity) ranging from 0.3 to 1.7 were studied. Most of the strain recovery was attributable to interfacial tension, and could be well-described by just two parameters: the ultimate recovery and a single retardation time. Both these parameters were found to increase with the capillary number of the drops prior to cessation of shear. For blends that had reached steady shear conditions, the ultimate recovery decreased with increasing viscosity ratio, whereas the retardation time increased with increasing viscosity ratio. The retardation time was well-predicted, but the ultimate recovery was over-predicted by a linear viscoelastic model developed previously by Vinckier et al. (Rheol Acta 38:65–72, 1999).     

Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer

The effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid saturated porous layer is investigated by a linear stability analysis. Modified Darcy–Maxwell model is used to describe the fluid motion. The regular perturbation method based on the small amplitude of modulation is employed to compute the critical Rayleigh number and the corresponding wavenumber. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the viscoelastic parameter, Darcy–Prandtl number, normalized porosity, and the frequency of modulation. It is found that the low frequency symmetric thermal modulation is destabilizing while moderate and high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing while the system becomes unstable for large values of Darcy–Prandtl number and for small frequencies. It is shown that in general the gravity modulation produces a stabilizing effect on the onset of convection for moderate and high frequency. The small frequency gravity modulation is found to have destabilizing effect on the stability of the system.     

Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia

The vibration problem of a viscoelastic cylindrical shell is studied in a geometrically nonlinear formulation using the refined Timoshenko theory. The problem is solved by the Bubnov–Galerkin procedure combined with a numerical method based on quadrature formulas. The choice of relaxation kernels is substantiated for solving dynamic problems of viscoelastic systems. The numerical convergence of the Bubnov–Galerkin procedure is examined. The effect of viscoelastic properties of the material on the response of the cylindrical shell is discussed. The results obtained by various theories are compared.     

Stability of plane Poiseuille flow of slightly viscoelastic fluids

Summary The title subject has been examined by the author in a series of papers (Cousins, 1970, 1972a, b), and the assumptions and principal results of those papers are discussed here. The work is motivated by the phenomenon evinced in fluid flow situations, of turbulent drag reduction by certain polymer additives. From a survey of experimental work it is clear that molecular elongation plays an important role in reducing drag by suppressing transverse motions. This effect may be interpreted as a normal stress effect in a continuum theory. A second-order fluid, which is a simple model exhibiting such a property, is used in a linear analysis of disturbances to planePoiseuille flow. Unlike theNewtonin case Squire's theorem is not valid (Lockett, 1969a) and a three-dimensional analysis is required. The viscoelastic terms are in general destabilising. Under certain conditions the first growing disturbance will propagate at an angle to the basic flow, giving a longitudinal vortex structure close to the channel boundaries not present at the onset of instability in aNewtonian fluid. The analysis is extended to finite-amplitude disturbances by introducing a time-dependent amplitude, but calculations are here confined to the simpler two-dimensional case. Disturbances which would decay under linear theory may in fact grow provided the initial amplitude is sufficiently large. A threshold amplitude for instability is found as a function ofReynolds number. The viscoelastic terms are again found to be destabilising. Finally, a further viscoelastic property, that of stress relaxation, is introduced through an integral representation of the stress. A linear analysis is developed and stress relaxation is also shown to be a destabilising influence.With 6 figures     

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter \(\Lambda _2 \) portrays stabilizing influence on the system while increasing stress relaxation parameter \(\Lambda _1\) displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.     

Rotational flow of Oldroyd-B nanofluid subject to Cattaneo-Christov double diffusion theory

A nanofluid is composed of a base fluid component and nanoparticles, in which the nanoparticles are dispersed in the base fluid. The addition of nanoparticles into a base fluid can remarkably improve the thermal conductivity of the nanofluid, and such an increment of thermal conductivity can play an important role in improving the heat transfer rate of the base fluid. Further, the dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications...     

Influence of the viscoelastic properties of a composite on the stress distribution around an elliptic hole in a plate

The time variation in the stresses around an elliptic hole in a composite plate is studied. Solutions that characterize the effect of the time dependence of the relaxation moduli of the composite components on stresses are obtained. The solutions in the time domain are obtained from the elastic–viscoelastic analogy and the corresponding elastic solutions for the effective moduli of the composite and the stress field around an elliptic hole in an anisotropic plate. The inverse Laplace transformation is carried out by an effective numerical method     

Concentration dependence of the linear viscoelastic properties of particle suspensions

The response under small amplitude oscillatory deformations of a suspension of non-Brownian spheres dispersed in a viscoelastic fluid is investigated. The correspondence principle of linear viscoelasticity is used to derive a simple constitutive model from a model for a suspension in a Newtonian liquid. The theory predicts that for a specific particulate system the concentration dependence of the viscoelastic properties should collapse to a single master curve when the values are normalized with those of the carrier fluid alone. Measurements with the micro-Fourier rheometer using oscillatory squeeze flow are carried out on two suspensions of 60 and 80 μm sized particles dispersed in polymeric fluid and in silicon oil, and the master curve is verified. Received: 27 April 1999/Accepted: 15 October 1999     

Effect of purely elastic flow instability on molecular conformation and drag

Linear stability analysis has shown that viscoelastic creeping flow of an Oldroyd-B liquid through a sinusoidal channel is unstable to stationary, wall-localized and short wavelength perturbations [B. Sadanandan, R. Sureshkumar, Global linear stability analysis of non-separated viscoelastic flow through a periodically constricted channel, J. Non-Newtonian Fluid Mech. 122 (2004) 55]. In this work, time-dependent simulations are performed to determine the nonlinear evolution of finite amplitude disturbances in the post-critical flow regime. It is shown that a nonlinear transition, which is facilitated by a supercritical pitchfork bifurcation, establishes a finite amplitude state (FAS) in which the average polymer stretch is highly modulated. The maximum normal stress, observed at the channel nip, can increase by up to approximately 100% when the Weissenberg number, defined as the ratio of the fluid relaxation time to an inverse characteristic shear rate, is increased by only 10% beyond its critical value. This is attributed to the amplification of configurational perturbations by the base flow shear rate, which attains its maximum at the channel nip. The effect of finite chain extensibility on the critical condition and nonlinear instability is investigated using the FENE-CR model. The stabilizing effect of finite extensibility can be expressed through a renormalization of the Weissenberg number by accounting for the screening effect of the nonlinear force law on the transmission of configurational perturbations to polymeric stress. The principal features of the FAS are qualitatively model-independent. The FAS exhibits a small, but numerically perceptible increase in the friction factor as compared to the base flow. The implication of the findings on the experimentally observed flow resistance enhancement phenomenon in viscoelastic creeping flows through converging/diverging geometries is discussed.     

Stability of a viscoelastic plate in fluid flow

The stability of an infinite viscoelastic plate on an elastic foundation in a viscous incompressible flow is studied. The Navier-Stokes system is linearized for an exponential velocity profile. The problem is reduced by a Fourier-Laplace transform to a system of ordinary differential equations, whose solution is found in the form of convergent series. The roots of the dispersion relation that characterize the stability of the system are found numerically. The effect of the viscosities of the fluid and the plate on the stability of the waves propagating upstream and downstream is studied. The results are compared with available data on the stability of a viscoelastic plate in an ideal fluid flow. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 66–74, July–August, 2006.     

A differential model for the rheological properties of concentrated suspensions with weakly viscoelastic matrices

A model for the rheological properties of a concentrated suspension in weakly viscoelastic fluid matrices is proposed. The model is derived according to the Roscoe differential procedure described in 1952. The analytical results produced recently by Greco et al. (J Non-Newton Fluid Mech 147:1–10, 2007) and Housiadas and Tanner (J Non-Newton Fluid Mech 162:88–92, 2009) for dilute suspensions of neutrally buoyant, non-Brownian rigid spheres in weakly viscoelastic matrix fluids are the key results which are used as a base to predict the properties of concentrated suspensions. The results are compared with the few available experimental data from the literature, showing promising trends for the viscometric properties of the suspensions. In particular, one sees the rapidly increasing value of −N₂/N₁ as concentration increases.     

Nanoscale Poiseuille flow and effects of modified Lennard–Jones potential function

Numerical simulation of Poiseuille flow of liquid Argon in a nanochannel using the non-equilibrium molecular dynamics simulation (NEMD) is performed. The nanochannel is a three-dimensional rectangular prism geometry where the concerned numbers of Argon atoms are 2,700, 2,550 and 2,400 at 102, 108 and 120 K. Poiseuille flow is simulated by embedding the fluid particles in a uniform force field. An external driving force, ranging from 1 to 11 PN (Pico Newton), is applied along the flow direction to inlet fluid particles during the simulation. To obtain a more uniform temperature distribution across the channel, local thermostating near the wall are used. Also, the effect of other mixing rules (Lorenthz–Berthelot and Waldman–Kugler rules) on the interface structure are examined by comparing the density profiles near the liquid/solid interfaces for wall temperatures 108 and 133 K for an external force of 7 PN. Using Kong and Waldman–Kugler rules, the molecules near the solid walls were more randomly distributed compared to Lorenthz–Berthelot rule. These mean that the attraction between solid–fluid atoms was weakened by using Kong rule and Waldman–Kugler rule rather than the Lorenthz–Berthelot rule. Also, results show that the mean axial velocity has symmetrical distribution near the channel centerline and an increase in external driving force can increase maximum and average velocity values of fluid. Furthermore, the slip length and slip velocity are functions of the driving forces and they show an arising trend with an increase in inlet driving force and no slip boundary condition is satisfied at very low external force (<1 PN).